A Self-adjointness Criterion for the Schrodinger Operator with Infinitely Many Point Interactions and Its Application to Random Operators

We prove the Schrodinger operator with infinitely many point interactions in R-d (d = 1, 2, 3) is self-adjoint if the support Gamma of the interactions is decomposed into infinitely many bounded subsets {Gamma(j)}(j) such that inf(j not equal k) dist(Gamma(j), Gamma(k)) > 0. Using this fact, we prove the self-adjointness of the Schrodinger operator with point interactions on a random perturbation of a lattice or on the Poisson configuration. We also determine the spectrum of the Schrodinger operators with random point interactions of Poisson-Anderson type.